How can I do an animation of an object approaching the camera?
From your description, path animation comes to mind. Here is a simple example. Please clarify if this is not what you have in mind.



To set this up, add a curve object to your scene. Shift + ACurve> Bezier (or any other curve object).

Next, edit the curve to fit the path you want your object to follow as it moves toward the camera. Here is what my example above looks like from side view. 



Finally, parent your object to the curve with Ctrl + P>Follow Path . By default, this will give you 100 frames. If you need to make adjustments, you can do so in the curve properties in the properties panel.



From here, Alt+A to play the animation, and  then if needed, make any necessary adjustments.